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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
{\bf x}^{\prime}={\bf A}\,{\bf x}, \quad {\bf A} \mathrm{~is ~real}.\tag{6.4.1}
\end{equation}
</div>
<p class="continuation">The corresponding eigenvalue problem is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
({\bf A}-r{\bf I}) \vec{\xi}={\bf 0}.\tag{6.4.2}
\end{equation}
</div>
<p class="continuation">Suppose that there is a <span class="process-math">\(k-\)</span> repeated eigenvalues <span class="process-math">\(r=\rho\text{,}\)</span> i.e. <span class="process-math">\(\rho\)</span> is the <span class="process-math">\(k-\)</span> fold root of</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
|{\bf A}-r {\bf I}|=0.\tag{6.4.3}
\end{equation}
</div>
<p class="continuation">For the corresponding eigenvectors, there are two possibilities:(i) There are <span class="process-math">\(k-\)</span> linear independent eigenvectors <span class="process-math">\(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \cdots \vec{\xi}^{(k)}\text{.}\)</span> In this case, we have <span class="process-math">\(k-\)</span> linear independent solutions</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{(1)}=\vec{\xi}^{(1)} e^{\rho t}, \quad
{\bf x}^{(2)}=\vec{\xi}^{(2)} e^{\rho t}, \cdots
{\bf x}^{(k)}=\vec{\xi}^{(k)} e^{\rho t}.
\end{equation*}
</div>
<p class="continuation">(ii) There are less than <span class="process-math">\(k\)</span> linear independent eigenvectors. In this case, we have to find other solutions.</p>
<span class="incontext"><a href="sec6_4.html#p-271" class="internal">in-context</a></span>
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